Transcendence conjecture for the Upsilon constants

For $n\geq -1$, define the Upsilon constants by

$$\Upsilon_n = \int_0^{+\infty} \frac{1}{t^n} \left (\frac{1}{1-e^{-t}}- \sum_{k=-1}^{n} b_k t^{k} \right ) \frac{e^{-t} \, dt}{t}.$$

The Stirling-Ramanujan constants satisfy

$$S_n= (-1)^{n+1} n! (\Upsilon_n -r_n).$$

Upsilon transcendence conjecture. Upsilon constants $\Upsilon_n$ are transcendental.

The source states that, for $n\geq -1$, this conjecture is equivalent to the transcendence conjecture for the Stirling-Ramanujan constants, up to the displayed rational-affine relation.