The conjectured extension of the Burnside–Stirling–Gautschi bound

Let $\widehat{m}_{1/2}(x,\alpha)$ denote the logarithmic error term for the shifted interpolation parameter $\alpha$, let $\widehat{D}_{1/2}$ be its stated domain, and let $\overline{b_{SG}^\star}$ be the upper Burnside–Stirling–Gautschi bound. The conjectured extension of the Burnside–Stirling–Gautschi bound. For all $(x,\alpha)\in\widehat{D}_{1/2}$,

$$\widehat{m}_{1/2}(x,\alpha)\leq\overline{b_{SG}^\star}(x+\alpha).$$

Equivalently, the restriction $x\geq 1$ appearing in the preceding corollary should not be necessary. The claim is motivated by the numerical comparisons in the paper, and no resolution is supplied in the given text.