Solid-algebra persistence conjecture for cohomological dimension

Let $R$ be a Noetherian domain, let $S$ be a solid $R$-algebra, and let $I$ be an ideal of $R$. Solid-algebra persistence conjecture. One has

$$\operatorname{cd}(I,R)=\operatorname{cd}(IS,S).$$

The conjecture asks whether cohomological dimension is preserved when passing from a Noetherian domain to a solid algebra. The surrounding discussion notes that cohomological dimension can decrease under arbitrary ring homomorphisms, while equality is known in some cases such as direct-summand extensions; the asserted equality for all solid algebras remains open in the supplied text.