Let $B_n$ denote the Bernoulli numbers and let $\phi=\frac{1+\sqrt{5}}2$ be the golden ratio.

I encountered the following infinite sum and would like to ask:

Question. Is this true? If so, any proof? $$\sum_{k=1}^{\infty}\sum_{j=k}^{2k}\binom{k}{j-k}\frac{B_{j+1}}{j+1} =\frac{2\,\log\phi}{1-2\phi}.$$

Caveat. Do not try reversing summations, it diverges!