Recursive decomposition coefficient inequality for real-rooted f-polynomials

Let $f(t)=1+\displaystyle{\sum_{i=1}^d} f_{i-1}t^i\in\mathbb{N}[t]$ be a polynomial whose zeros are all real. Suppose that its recursive decomposition is $f(t)=g(t)+th(t)$, where

$$g(t)=1+\displaystyle{\sum_{i=1}^d}g_it^i,\qquad h(t)=1+\displaystyle{\sum_{i=1}^{d-1}}h_it^i.$$

Recursive decomposition inequality. For every relevant index $i$, one has $h_i\leq g_i$.

The recursive decomposition arises from the binomial expansions of the coefficients of an $f$-polynomial and is intended to yield a positive answer to the question referenced in the source. The supplied text does not state whether this conjecture has been resolved.