Existence of p-bases for higher Frobenius sandwiches

Let ${\mathbb K}$ be a perfect field of characteristic $p$, and let $R\geq S\geq R^q$ be a higher Frobenius sandwich of commutative local regular ${\mathbb K}$-algebras, where $q=p^s$ for some natural number $s$. A $p^\bullet$-basis of $R$ over $S$ is a sequence of elements $a_1,\ldots,a_n\in R$ together with natural numbers $k_1,\ldots,k_n$ such that the monomials

$$a_1^{m_1}\cdots a_n^{m_n},\qquad 0\leq m_i<p^{k_i},$$

form an $S$-basis of $R$.

Existence conjecture for higher Frobenius sandwiches. There should exist a $p^\bullet$-basis of $R$ over $S$.

Such a basis would provide a useful description of the structure of regular local rings in higher Frobenius sandwiches and support a broader generic-smoothness theory for inseparable maps. The source gives no resolution, so the conjecture remains open.