Generalized Frobenius bound for h-regular pairs

Let $(d_1,\ldots,d_c;a_0,\ldots,a_n)$ be an $h$-regular pair, and assume $c\leq n$ and $a_i\nmid h$ for every $i$. Let $F^h(a_0,\ldots,a_n)$ denote the $\frac{1}{h}$-Frobenius number, namely the largest multiple of $h$ not representable as a nonnegative integer combination of the weights.

Generalized Frobenius conjecture. Then

$$\delta(d;a)\geq F^h(a_0,\ldots,a_n).$$

For $h=1$, this specializes to the regular-pair Frobenius conjecture attributed to Tasin. The source introduces this as the generalization to $h$-regular pairs; its resolution is not stated, so the conjecture remains open.