  We confirm, in dimension two, Blocki's conjectures on sharp lower bounds for
Bergman kernels of tube domains. To that end, we verify a broader class of
$L^p$-Mahler conjectures due to Berndtsson and the authors, where $p=1$ are
Blocki's conjecture, and $p=\infty$ are Mahler's conjectures. The proofs are
technically challenging as the $L^p$-Mahler volume is considerably harder to
deal with analytically compared to Mahler's volume, and furthermore duality is
lost. In addition, unlike in the classical Mahler setting, the non-symmetric
setting is considerably more involved than the symmetric one. The proofs
involve studying the effect of Mahler's classical sliding of vertices on
two-dimensional polytopes on the $L^p$-polar body (no longer a polytope). Some
arguments are inspired by works of Campi--Gronchi and Meyer--Reisner on volumes
of classical polar bodies of shadow systems. In passing, we also explore how
Mahler's sliding affects the isotropic constant. This leads to an elementary
proof of Bourgain's strong hyperplane conjectures in dimension two, originally
due to Bisztriczky--B\"or\"oczky, Campi--Colesanti--Gronchi and Meckes.
Specifically, we show that, as a function of the sliding parameter, the
isotropic constant raised to an appropriate power is a convex quadratic
polynomial.
