Lynch's cohomological-dimension conjecture for local rings

Let $(R,\mathfrak m)$ be a local ring and let $I$ be an ideal of $R$. For an ideal $I$, write $\operatorname{cd}(I,R)$ for its cohomological dimension, and let $H^c_I(R)$ denote the $c$-th local cohomology module of $R$ supported at $I$. Lynch's conjecture. If the cohomological dimension of $I$ is a positive integer $c$, then

$$\dim R/\operatorname{ann}_R H^c_I(R)=\dim R/H^0_I(R).$$

The conjecture is false: the first counterexamples are nonequidimensional local rings with $\dim R\geqslant 5$.