On minimal Type IIB $AdS_6$ solutions with commuting 7-branes

We construct Type IIB supergravity solutions with geometry $AdS_6\times S^2$ warped over a disc with two boundary points where 5-branes emerge and punctures with 7-brane monodromy. They describe $(p,q)$ 5-brane junctions with two groups of like-charged external 5-branes that are unconstrained by the $s$-rule and an additional group of constrained 5-branes. The dual 5d SCFTs include various theories discussed previously in the literature. We match SCFT operators with scaling dimension of $\mathcal O(N)$ with their representation in supergravity to support the proposed dualities.