Genericity of shadowing for locally connected continua

Let $X$ be a locally connected continuum. Let $\mathcal{T}(X)$ denote the class of maps with shadowing, and let $\mathcal{C}(X)$ and $\mathcal{S}(X)$ denote the spaces of continuous and surjective self-maps of $X$, respectively.

Genericity conjecture. $\mathcal{T}(X)$ is generic in $\mathcal{C}(X)$ and $\mathcal{S}(X)$.

This conjecture concerns the prevalence of shadowing among continuous and surjective self-maps of locally connected continua. It is motivated by genericity results for various classes of locally connected continua and by the preceding question about whether every locally connected one-dimensional continuum is a graphite; the conjecture is presented as open in the source.