Let $X$ be an infinite dimensional Banach space and $T:X\rightarrow X$ be a bounded linear operator. If $T$ is invertible and $\parallel T\parallel_{e}=\parallel T\parallel$, is it true that(or when is it true that) $\parallel T^{-1}\parallel=\parallel T^{-1}\parallel_{e}$ ? Here $\parallel .\parallel_{e}$ denotes the essential norm.

Let $X$ be an infinite dimensional Banach space and $T:X\rightarrow X$ be a bounded linear operator. If $T$ is invertible and $\lVert T\rVert_e=\lVert T\rVert$, is it true that  (or when is it true that) $\lVert T^{-1}\rVert=\lVert T^{-1}\rVert_e$? Here, $\lVert\cdot\rVert_e$ denotes the essential norm.