Universality conjecture for the strongly co-Stone hull of a residuated lattice

Let $A$ be a residuated lattice, let $\tilde{A}$ be its strongly co-Stone hull, and let $\epsilon:A\to\tilde{A}$ be the canonical morphism. For a subset $X\subseteq A$, write $X^{\top}$ for its associated co-annihilator; similarly, for subsets of another residuated lattice, use the same notation. A residuated-lattice morphism is a map preserving the residuated-lattice structure. Universality property of the strongly co-Stone hull. For every strongly co-Stone residuated lattice $A_{1}$ and every residuated-lattice morphism $f:A\to A_{1}$ satisfying

$$f(X^{\top})=f(X)^{\top}$$

for every $X\subseteq A$, there exists a unique residuated-lattice morphism $\overline{f}:\tilde{A}\to A_{1}$ such that

$$\overline{f}\circ\epsilon=f.$$

This asserts that $\tilde{A}$ is universal among strongly co-Stone residuated lattices receiving a morphism from $A$ that preserves co-annihilators. The source provides no evidence resolving this statement, so its status is unclear.