In the case of a weld, initial grain growth occurs as atoms attach themselves to preexisting grains at the boundary between the liquid metal and the still solid HAZ.  As the grains grow, there is less liquid until the weld is completely solidified and there is no more liquid. The faster the liquid cools, the more crystals nucleate and form in the liquid instead of simply attaching to the preexisting grains. The faster the puddle cools, the smaller the grains are, but more of them form. So, slow cooling results in fewer, but larger grains. Faster cooling results in more grains, but they are smaller.

Grain size reduction can occur if the larger grains are strained such as is the case with strain hardening. They break into smaller grains as the metal is raised to a temperature above the recovery temperature. There is always strain involved when the weld and adjacent base metals cool and contract. The uneven distribution of thermal gradients result in residual stresses which causes the grains to strain.

Large grains can become larger by absorbing much smaller grains if it will reduce the overall energy, but there has to be a driving force for this to occur. Temperature is often the driving force required. There is a limit to how large the grains can become because they will not absorb grains that are nearly the same size unless near liquidus temperatures are attained. This isn't the case unless the discussion is limited to the "mushy" zone adjacent to the liquid weld puddle. The time that the weld puddle is in the necessary temperature range is relatively short, so the grain growth is very limited unless extreme welding conditions are encountered.

In the case of welding, the initial formation of grains occurs as the weld solidifies. The size of the grains increases as the atoms in the liquid attach themselves to the preexisting grains. The size of the grain is dependent on the length of time the weld is in the liquid state. In the HAZ, the grains can coarsen or grow if they base metal is held at a temperature above the recrystallization  temperature and just below the melting temperature for relatively long periods of time as the larger grains devour the smaller grains. This is not the expected case if the welding technique was to employ multiple stringer beads and the interpass temperature was controlled.

However, if SAW was used, with slow travel speeds, high heat input, and very thick weld beads were deposited, the very slow cooling rates could in fact cause grain coarsening. Likewise, high interpass temperatures result in increasing the time at temperature of the weld and heat affected zones which could result in grains coarsening in the weld and some grain growth in the HAZ. The time  at temperature of concern are well into the austenizing temperatures for steel alloys. A temperature rise of a few hundred degrees isn't sufficient to cause recovery or recrystallization. Likewise the time required to initiate recrystallization is usually longer than that experienced when stringer beads are deposited.

At least, that's how I understand the situation.

Best regards - Al

If I had to guess what the question was getting at, since it does say "welds underneath", my answer would be to get smaller since there is a lot of grain refining that happens when welding over subsequent weld beads in ordinary steels.

PD,
I understand your dilemma. This situation seems to exist in many testing and certification environments. AWS is no excpetion. And as Al and Greg made clear it can be very complicated. It is, very complicated. The proper answer to your multiple choice question IMO would essentially be all three. Bigger, smaller, and no change, depending upon where in the cycle the particular HAZ location is, the temps involved the base metals, etc.
But of course, testing enviroments think they are keeping it simple and basic (or when they really get creative make a annoying attempt to try an make you think-which is actually synonymous with thinking like them)when what they are actually doing is forcing the tester to get inside the heads of the person that formed the question because the correct and most accurate answer is not available. So you are left with a guess. Or, you take a seminar so that you can memorize the pat answer like a monkey pullin a lever.