A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space

We present a 2-dimensional Peano continuum $\mathbb{T}\subseteq {\mathbb{R}}^3$ with the following properties: (1) There is a universal covering projection $q:\bar{\mathbb{T}}\to \mathbb{T}$ with uncountable fundamental group ${\pi }_1(\bar{\mathbb{T}})$; (2) For every $1≠[\bar{\alpha }]\in {\pi }_1(\bar{\mathbb{T}},\ast )$, there is a covering projection $r:(E,e)\to (\bar{\mathbb{T}},\ast )$ such that $[\bar{\alpha }]\in ̸r_{\mathrm{\#}}{\pi }_1(E,e)$; (3) There is no universal covering projection $r:E\to \bar{\mathbb{T}}$; (4) The universal object $p:\tilde{\mathbb{T}}\to \mathbb{T}$ in the category of fibrations with unique path lifting (and path-connected total space) over $\mathbb{T}$ has trivial fundamental group ${\pi }_1(\tilde{\mathbb{T}})=1$; (5) $p:\tilde{\mathbb{T}}\to \mathbb{T}$ is not a path component of an inverse limit of covering projections over $\mathbb{T}$.