= Solution
The scalar <logistic loss> $u\mapsto-u+\log(1+e^u)$ is strictly convex because its second derivative is $e^u/(1+e^u)^2>0$. If $\widehat\beta$ and $\widetilde\beta$ are minimizers but $X\widehat\beta\ne X\widetilde\beta$, strict convexity of the loss as a function of the fitted vector and convexity of the <L1 norm> make the objective at their midpoint strictly smaller than the common minimum. This contradiction proves that
$$
\boxed{X\widehat\beta=X\widetilde\beta}
$$
for every pair of solutions.
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