Universal partial sums of Taylor series as functions of the centre of expansion

V. Nestoridis conjectured that if $\mathrm{\Omega }$ is a simply connected subset of $\mathbb{C}$ that does not contain $0$ and $S(\mathrm{\Omega })$ is the set of all functions $f\in \mathcal{H}(\mathrm{\Omega })$ with the property that the set $\left{T_N(f)(z)\coloneqq\sum_{n=0}^N\dfrac{f^{(n)}(z)}{n!} (-z)^n : N = 0,1,2,\dots \right}$isdensein$$\mathcal{H}$(\Omega)$,then$S(\Omega)$ is a dense $G_{\delta }$ set in $\mathcal{H}(\mathrm{\Omega })$. We answer the conjecture in the affirmative in the special case where $\mathrm{\Omega }$ is an open disc $D(z_0,r)$ that does not contain $0$.