Regularity and Gröbner basis conjecture for powers of binary-form ideals

Let $U$ be a two-dimensional vector space, let $a,b$ be positive integers, and write $S=\operatorname{Sym}(\operatorname{Sym}^b U)$. Let $I_{a,b}=\langle\operatorname{Sym}^d U\rangle$ be the primary ideal generated by the indicated copy of $\operatorname{Sym}^d U$ in $S$. Regularity and Gröbner basis conjecture. For $1\leq j\leq b$, one has

$$\operatorname{reg}(I_{a,b}^j)=\left\lfloor\frac{b+j+1}{2}\right\rfloor a-\left\lfloor\frac{b-j+1}{2}\right\rfloor,$$

and the minimal generators of $I_{a,b}^j$ form a Gröbner basis. This conjecture concerns the powers of the base ideal associated with the linear series of binary forms and would provide information about their regularity and Rees-algebra-related syzygies; the supplied text does not state whether it has been resolved.