Forcing-independence conjecture for the large deviations principle

Consider the system with forcing $f\in L^2(0,T;H_0)$, and suppose that a large deviations principle holds for at least one such forcing. The forcing-independence conjecture asserts that the large deviations principle is independent of the choice of forcing: if it holds for at least one forcing $f\in L^2(0,T;H_0)$, then it holds for every other forcing in that space.

Forcing-independence conjecture. The large deviations principle for the system holds independently of the choice of the forcing $f\in L^2(0,T;H_0)$; equivalently, if an LDP holds for at least one such forcing, then it holds for every other.

A positive answer would yield a full family of large deviations principles for systems with arbitrary forcing. The source motivates the claim through the contraction principle and notes that transferring an LDP between different forcings appears to require convergence information associated with the strong Kato condition; no resolution is supplied.