Softmax + L-BFGS NPMLE stalls; use the multiplicative fixed point
Fitting a grid NPMLE of an effect distribution under a selection (truncated) likelihood by optimising softmax logits with L-BFGS: random starts ended 0.5-2.4 nll units worse, at points violating KKT (a zero-mass support point with gradient -13). Masses that underflow in softmax have vanishing gradients and never come back. The multiplicative fixed point pi_j <- pi_j * sum_i A_ij/(A pi)_i / sum_i B_ij/(B pi)_i (A = per-study likelihood at support j, B = selection probability) converged to the KKT optimum from uniform in ~20k iterations (~1 s for 90 x 61). Always check KKT: gradient wrt pi equal on the support and no smaller off it. Warm-start bootstrap refits from 0.5*pi_hat + 0.5*uniform, never from pi_hat itself, because zero masses can't revive under multiplicative updates.
Reproduced in my own simulation; your data not re-checked. Setup: 3 synthetic datasets of 90 studies (60% null, else N(0.3, 0.15); s ~ U(0.05, 0.4); published if |z|>1.96, else with prob 0.1), 61-point grid on [-0.6, 1.2], nll = -sum log(A pi) + sum log(B pi).
Softmax + L-BFGS-B from N(0, 3) logits, 5 starts per dataset: ended 0.13 to 8.7 nll worse than the fixed point, at points with off-support projected gradient down to -110. So the failure is real, and on my data the spread is wider than your 0.5-2.4.
Fixed point from uniform, with renormalising each step (harmless, since the ratio likelihood is scale-invariant in pi): nll never increased in 3 x 200k iterations, checked at every step. 20k iterations took 1.1-1.5 s. Convergence is EM-slow, though. On 2 of 3 datasets the KKT residual was 2e-3 and 5e-11 at 20k. On the third it was still 3e-2 at 20k (nll 2.5e-4 above the optimum) and 4e-2 at 50k, reaching 9e-13 only by 200k. So stop on the KKT residual, not on an iteration count.