On Ж, and the fine-structure constant

Question

I'm trying to reproduce the results from a certain infamous paper that has been moving around the web for the last few days. The details are irrelevant.

This paper claims to have a closed-form expression for the so-called fine-structure constant $Ж:=\alpha^{-1}=137.035\dots$, to wit, $$ Ж=\lim_{\substack{n\to\infty\\j\to\infty}}2^{-2n}B^n_{k(j)}\tag{8.11} $$ where $k(0):=0$ and $k(j+1):=2^{k(j)}$, and $B^n_k$ are the Bernoulli polynomials.

A very naïve implementation of this formula is as follows:

Clear[k]
k[0] = 0;
k[j_?NumericQ] := 2^k[j - 1];
Table[N[2^(-2 n) BernoulliB[n, k[n]], 50], {n, 1, 10}]

which diverges (and actually overflows), suggesting that the limit does not exist. Is this correct, or is the apparent divergence just an issue of numerical instability? If so, what is the correct value for the limit above? Is $Ж=137.035\dots$ correct?

The paper offers alternative formulas, such as $$ Ж=\frac{\pi}{\gamma}Ч\tag{1.1} $$ where $\gamma$ is the Euler constant, and $$ Ч:=\frac12\lim_{n\to\infty}\sum_{j=1}^n2^{-j}\left(1-\int_{1/j}^j\log_2(x)\mathrm dx\right)\tag{7.1} $$

Unlike before, this alternative formula seems to converge, but not to the claimed value. Indeed,

π/(2 EulerGamma) NSum[2^-j (1 - Integrate[Log[2, x], {x, 1/j, j}]), {j, 1, ∞}]

evaluates to 0.16026, nowhere near $137$. Again, is this numerical calculation reliable, or is the disagreement due to numerical error? If so, what is the correct value of this sum?