We study prophet inequalities under intersections of $q$ partition matroids, where an online algorithm irrevocably selects elements with independent nonnegative values drawn from known distributions and revealed in an adversarial order. We prove an $Ω(q/\log q)$ lower bound on the competitive ratio. Together with the known $O(q)$ upper bounds, this resolves, up to a logarithmic factor, the optimal dependence on $q$, an open question posed by Correa, Cristi, Fielbaum, Pollner, and Weinberg (IPCO 2022) and Saxena, Velusamy, and Weinberg (ITCS 2023). Our construction also yields an $Ω(d/\log d)$ lower bound for $d$-single-minded auctions, where buyers request fixed bundles of at most $d$ unit-capacity items. Our construction and analysis build on the big-decisions-first framework of Rubinstein and Singla (STOC 2026). Previous lower bounds for this problem hold even for i.i.d. Bernoulli values, whereas our instance uses nonidentical distributions. For i.i.d. Bernoulli values, we prove a $q^{1-o(1)}$ lower bound, improving the previous $q^{1/2+Ω(1/\log\log q)}$ of Saxena, Velusamy, and Weinber (ITCS 2023). To this end, we construct $p^p$ pairwise qualitatively independent partitions into $p$ blocks on a ground set of only $p^{2+o(1)}$ points.

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