Negative-argument congruence conjecture for partial Stirling functions

Let $P_n(x)$ denote the partial Stirling function, let $\nu$ be the 2-adic valuation, and let $\operatorname{o}(n)=n/2^{\nu(n)}$ be the odd part of $n$. For integers $e$ and $\Delta$ in the range considered in the paper, and for $x<0$, the negative-argument congruence conjecture.

$$P_{2^e+\Delta}(x)\equiv\frac{2}{\operatorname{o}(2^e!)}P_\Delta(x)\pmod {2^{e-\operatorname{lg}([\Delta/2]-x)}}.$$

This would extend the corresponding congruence already obtained for $0\le x<\Delta$ and describe the limiting behavior of these functions at negative arguments.