Edit after a first interesting answer.

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.

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.

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

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.

Edit after a first interesting answer.

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.