Adin's nonnegativity conjecture for the cubical $g$-vector

Let $Q$ be a cubical $d$-polytope, meaning that every proper face of $Q$ is combinatorially a cube. Its cubical $h$-polynomial is written as

$$h^c(Q,t)=\sum_{i=0}^d h^c_i t^i,$$

and its cubical $g$-vector is $g^c(Q)=(g^c_0,\ldots,g^c_{\left\lfloor d/2\right\rfloor})$, where $g^c_0=h^c_0=2^{d-1}$ and $g^c_i=h^c_i-h^c_{i-1}$ for $1\leq i\leq\left\lfloor d/2\right\rfloor$. Adin's conjecture. For a cubical $d$-polytope, one has

$$g^c_i\geq 0\qquad (1\leq i\leq\left\lfloor d/2\right\rfloor).$$

This conjectures the nonnegativity of all nontrivial entries of the cubical $g$-vector, analogous to the classical $g$-theorem for simplicial polytopes. The supplied source does not indicate whether the conjecture has been resolved.