An entrywise inverse-norm conjecture and its sharp replacement

A $3\times3$ positive strictly diagonally dominant matrix refutes the conjectured entrywise inverse-norm inequality $\|J^{-1}\|_\infty\ge\|S^{-1}\|_\infty$ for real symmetric, entrywise positive, diagonally dominant $J$ with $J\le S=\alpha I+m\mathbf 1\mathbf 1^T$ entrywise: for $n=3$ and $m=\alpha=1$, the matrix with diagonal entries $2$ and off-diagonal entries $1/2$ has $\|J^{-1}\|_\infty=7/9<5/4=\|S^{-1}\|_\infty$. Under the stated entrywise upper bounds, the sharp inverse infinity-norm infimum is $1/(\alpha+m)$, rather than the value at $S=\alpha I+m\mathbf 1\mathbf 1^T$; strict positivity of all entries prevents attainment. The conclusion concerns entrywise ordering, not a stronger order also comparing diagonal-dominance margins.