Five-ball construction conjecture for $h$-vectors

A five-ball is a triangulated ball of dimension five, with $h$-vector written as $h=(1,h_1,h_2,h_3,h_4,h_5,0)$. Theorem's construction gives a collection of $h$-vectors described by its stated conditions. Five-ball construction conjecture. The vector $h$ is the $h$-vector of a five-ball if and only if there exists an integer $m>0$ such that

$$h=(1,h_1-m,h_2,h_3,h_4,h_5,0)$$

satisfies the conditions of the construction theorem. This is presented as an open conjecture; determining a conjectural description of the $h$-vectors of higher-dimensional balls remains an open problem.