The massive essential spectrum conjecture for singular integral operators on variable Lebesgue spaces

Let $\mathbb{T}$ be the unit circle, and let $a:\mathbb{T}\to\mathbb{C}$ be continuous on $\mathbb{T}\setminus\{1\}$ with $a(1-0)=0$ and $a(1+0)=1$. Let $P$ and $Q$ denote the operators occurring in $aP+Q$, and let $S$ be the Cauchy singular integral operator. For a measurable exponent $p:\mathbb{T}\to(1,\infty)$, define the essential spectrum by

$$\sigma_{\operatorname{ess}}(aP+Q):=\{\lambda\in\mathbb{C}:(aP+Q)-\lambda I\text{ is not Fredholm on }L^{p(\cdot)}(\mathbb{T})\}.$$

Massive essential spectrum conjecture. There exists a measurable function $p:\mathbb{T}\to(1,\infty)$ such that $S$ is bounded on $L^{p(\cdot)}(\mathbb{T})$ and $\sigma_{\operatorname{ess}}(aP+Q)$ has nonzero plane measure.

This claim predicts that, under sufficiently irregular variable exponents, boundedness of the singular integral can coexist with an essential spectrum of positive planar measure rather than the more familiar curve-like spectrum. The supplied passage presents it as the author's belief and gives no resolution.