The conjectured optimal bound for the logarithmic interpolation error

Let $\widehat{D}$ be the domain of pairs $(x,\alpha)$ used for the logarithmically interpolated factorial, and let

$$\widehat{\iota}(x,\alpha)=\int_1^{\infty}\frac{\phi_\alpha(t)}{x+t}\,\mathrm{d}t.$$

Define $\overline{b_G^\star}\colon\mathbb{R}_{\geq 0}\to\mathbb{R}$ by

$$\overline{b_G^\star}(y)=\frac{1}{8y+4}.$$

The conjectured optimal logarithmic interpolation bound. For all $(x,\alpha)\in\widehat{D}$,

$$\widehat{\iota}(x,\alpha)\leq\overline{b_G^\star}(x+\alpha).$$

The proposed bound has the correct asymptotic behaviour and is suggested as a possible improvement of the previously established bound; its validity remains open in the supplied text.