The interior-vertex lower-bound conjecture for grid pebbling distributions

Let $P$ be a solvable pebbling distribution on the grid. A vertex is interior if it does not lie on the boundary of the grid.

Interior-vertex lower-bound conjecture. For every interior vertex $X$,

$$v(X) \geq \frac{3}{2}.$$

The conjecture would improve the lower bound used in the paper from $\frac{4}{3}$ to $\frac{3}{2}$ for vertices away from the boundary, potentially yielding a stronger lower bound on the pebbling number of the grid. The source presents it as an open problem motivated by the observation about $2$-reachable vertices.