Polynomial bound relating homological degree and regularity

Let $A$ be a standard graded algebra of dimension $d$, and let $\operatorname{hdeg}(A)$ and $\operatorname{reg}(A)$ denote its homological degree and Castelnuovo–Mumford regularity.

Homological-degree regularity conjecture. There exists a polynomial $f(t)$ of degree $d$ such that

$$\operatorname{hdeg}(A)<f(\operatorname{reg}(A)).$$

The source notes that this holds for one known homological-degree function and asks for such a polynomial bound in general; no resolution is given.