Here is the proof of the Conjecture 1. The idea of the proof came from the upper answer by Danny Ruberman and by private communication with Semen Podkorytov. I am very grateful to them!

To prove this Conjecture we need some auxiliary observations. Let $M^{n+k}$ be a connected TOP $(n+k)$-manifold and $N^n\subset M^{n+k}$ be a closed connected TOP submanifold of dimension $n$. Both manifolds $N^n$ and $M^{n+k}$ are assumed to be of a finite homotopy type, but they may be nonorientable, and the inclusion $N^n\hookrightarrow M^{n+k}$ may be wild (non locally flat).

The following definition-proposition can be found in the classical textbook by A.Dold (Lectures on algebraic topology, 2ed., Chapter VIII, Corollary 11.20). Let the coefficient ring be $\mathbb{Z}_2:=\mathbb{Z}/2\mathbb{Z}$.

**Definition-Proposition.** *The cohomology group $H^i(M,M-N)$ is zero for $0\le i \le k-1$. The group $H^k(M,M-N)$ consists only one nonzero element $u_{M,N}$, which is called the Thom class of the pair $(M,N)$. Suppose $r\colon M' \to N$ is a retraction of an open domain $N\subset M' \subset M$. Then the composition
$$
H^q(N) \xrightarrow{r^*} H^q(M') \xrightarrow{\smile u_{M',N}} H^{q+k} (M',M' - N) \stackrel{\mathrm{exc}}{\cong} H^{q+k} (M,M-N)
$$
is an isomorphism for all $0\le q \le n$. Moreover, this isomorphism is independent of the choice of the domain $M'$ and a retraction $r\colon M' \to N$. This mapping is called the Thom isomorphism.*

Due to this fact we can give the following definition of Stiefel-Whitney classes of a connected topological manifold $M^n$ (closed or open and of a finite homotopy type).

**Definition 1.** *Let us consider the pair $M^n \stackrel{\Delta}{\hookrightarrow} M^n\times M^n$. By $\phi_{M}$ we will denote the Thom isomorphism $\phi_{M}\colon H^*(M) \to H^{*+n} (M\times M, M\times M - \Delta(M))$. We define the Stiefel-Whitney classes $w_k(M)$ by the formula $w_k(M) := \phi_{M}^{-1} \mathrm{Sq}^k \phi_{M} (1), 1\le k \le n$. Here, $\phi_{M}(1)$ is equal to the Thom class $u_{M\times M, \Delta(M)}$.*

Now, suppose that a connected topological manifold $M^n$ (closed or open and of a finite homotopy type) is smoothable and $\Sigma$ is a smooth structure on $M^n$. Then $\Sigma\times \Sigma$ is a corresponding smooth structure on $M^n\times M^n$, and $\Delta(M^n)\subset M^n\times M^n$ is a smooth submanifold. Moreover, the normal vector bundle $\xi_{M}$ of the submanifold $\Delta(M^n)\subset M^n\times M^n$ is isomorphic to the tangent bundle $\tau_{M}$ of $M^n$.

Let us fix some Riemannian metric $g_{M\times M}$ on the manifold $M^n\times M^n$. Then we can take some small tubular neighborhood $E_{M}$ of $\Delta(M^n)\subset M^n\times M^n$ and the corresponding retraction $r_{M}\colon E_{M} \to \Delta(M)$. The retraction $r_{M}$ is a strong deformation retraction. Therefore, $r_{M}^*\colon H^*(\Delta(M))\to H^*(E_{M})$ is an isomorphism.

Now, it is evident, that we are in the standard situation of vector bundles and their Stiefel-Whitney characteristic classes. We have the standard *Thom class* $u_{M}\in H^n(E_{M}, E_{M} - \Delta(M))$ of the vector bundle $\xi_{M}\cong \tau_{M}$, and the standard *Thom isomorphism* $\phi_{M}\colon H^*(M) \to H^{*+n} (E_{M}, E_{M} - \Delta(M))$, which is defined by the composition
$$
H^q(M) = H^q(\Delta(M)) \xrightarrow{r_{M}^*} H^q(E_{M}) \xrightarrow{\smile u_{M}} H^{q+n} (E_{M}, E_{M} - \Delta(M)), \ 0\le q\le n.
$$

Finally, we get the Stiefel-Whitney classes $w_k(M)$ of the tangent bundle $\tau_{M}$ by the *Thom formula* $w_k(M) := \phi_{M}^{-1} \mathrm{Sq}^k \phi_{M} (1), 1\le k \le n$.
But, due to the Definition-Proposition and the Definition 1, it is clear, that the Thom class, the Thom isomorphism and the Stiefel-Whitney classes obtained from the smooth structure $\Sigma$ on $M^n$ coincide with the *topological* Thom class, Thom isomorphism and Stiefel-Whitney classes of $M^n$.

Evidently, these observations prove the Conjecture 1.