Continuous bijection between two homotopy equivalent $\Delta$-generated spaces

Question

EDIT:

1. First edit after an interesting answer.
2. $(S,\mathcal{T}_1)$ and $(S,\mathcal{T}_2)$ are homotopy equivalent to the same Quillen cofibrant space.

Let $S$ be a set with two topologies $\mathcal{T}_1$ and $\mathcal{T}_2$ such that the identity map induces a continuous map $(S,\mathcal{T}_1)\to (S,\mathcal{T}_2)$ which is also a homotopy equivalence. I suppose these two topological spaces $\Delta$-generated.

Is it enough to conclude that $\mathcal{T}_1=\mathcal{T}_2$ ?

Motivation: I need to prove that some continuous bijection is a homeomorphism. Since the spaces are $\Delta$-generated, it suffices to prove that every continuous map from $[0,1]$ to $(S,\mathcal{T}_2)$ is a continuous map from $[0,1]$ to $(S,\mathcal{T}_1)$. I would like to use in some way the homotopy equivalence.

Answer 1

Let $X = S^1$ and $X^\delta$ its discretisation. Then the identity map $\iota: X^\delta \to X$ is a continuous bijection of $\Delta$-generated spaces, although not a homotopy equivalence. But the map of unreduced cones $$C(\iota): C(X^\delta) \to C(X)$$ is a homotopy equivalence and a continuous bijection, and both spaces are $\Delta$-generated.

Answer 2

[answer is to the original question before edit which posited only the existence of a homotopy equivalence rather than the identity map being a homotopy equivalence]

No. Let $T_2$ be the wedge of countably many circles and $T_1$ be the wedge of countably many circles and an interval.