Hahn–Banach extension conjecture for ultra pseudo-normed modules

Let $E$ be an ultra pseudo-normed $\widetilde{\mathbb K}$-module, let $E'$ denote its continuous linear functionals, and let $V$ be a submodule of $E$. For $\varphi\in V'$, the extension conjecture. There exists an extension $\psi\in E'$ such that

$$\|\psi\|=\|\varphi\|.$$

This is a Hahn–Banach-type extension assertion in the setting of ultra pseudo-normed modules. The source presents it as a conjecture and gives no resolution.