The shifted-sum conjecture for pure O-sequences

Let $h=(1,h_1,\dotsc,h_e)$ and $h'=(1,h'_1,\dotsc,h'_{e-1})$ be pure $O$-sequences. For a vector $H=(1,H_1,H_2,\dots,H_t)$, write $\Delta H=(1,H_1-1,H_2-H_1,\dots,H_t-H_{t-1})$ for its first difference. Assume that $(\Delta h')_i\leq(\Delta h)_i$ for all $i\leq\lceil e/2\rceil$ and that $h'_i\leq h_i$ for all $i\leq e-1$. Shifted-sum conjecture. The shifted sum

$$h”=(1,h_1+1,h_2+h'_1,\dotsc,h_e+h'_{e-1})$$

is also a pure $O$-sequence. The conjecture is designed to support an inductive approach to Stanley's conjecture via deletion and links of matroid complexes, and is proved in the paper for small socle degrees.