Conjecture on coincidence of nonlinear and constructed heat flow semigroups

Let $D$ be a measurable domain, let $P_{(D)}$ be the associated nonlinear Markov operator, and let $(T_t)$ be the heat flow semigroup defined by the paper's formula for $F:=P_{(D)}$. The semigroup is considered on the relevant space $L^2(D,\mathcal{H},h)$.

Heat-flow coincidence conjecture. The heat flow semigroup $(T_t)$ defined by the paper's formula for $F:=P_{(D)}$ coincides with the heat flow constructed by Sturm.

The conjecture asserts that the semigroup obtained from the paper's general nonlinear construction agrees with the previously constructed heat flow, without imposing the spectral bound condition used for the existence and uniqueness result cited earlier. Its status is not established by the supplied text.