See the result of Gerstenhaber-Rothaus, which says that if the abelianization of the word map has full rank, then the map $G\times G\to G\times G$ has non-zero degree. This is a necessary condition, as one can see if $G$ is abelian or has an abelian quotient (e.g. $U(n)$).
So this won't apply to $\{ab,ba\}$.

Once the map is non-zero degree, the pushforward of the Haar measure on $G\times G$ should be absolutely continuous with respect to Haar measure. This is because the map is also algebraic, and hence the preimage of points are smaller dimension, so the preimage of a set of measure $0$ will be measure $0$.

A theorem of Weyl implies that a compact subgroup of an algebraic group over $\mathbb{R}$ is an algebraic subgroup. Now we follow the argument in Barnea-Larsen, section 3.

*Barnea, Y.; Larsen, M.*, **Random generation in semisimple algebraic groups over local fields.**, J. Algebra 271, No. 1, 1-10 (2004). ZBL1049.20028.

Let's assume that $G$ is semisimple; I think that the general case can be reduced to this case. Since $G$ is compact, we may complexify to get a semisimple algebraic group $G^{\mathbb{C}}$ over $\mathbb{C}$. By Lemma 3.2, there is a countable set $\{X_0,X_1,\ldots\}$ of proper closed subvarieties such that if $\gamma\in G^{\mathbb{C}}- \cup_i X_i(\mathbb{C})$, then the Zariski closure of $\gamma$ is a maximal torus. Passing to $G=G^{\mathbb{R}}$, the real subgroup, we see that the same is true for $G$. Hence with probability $1$, any element $\gamma\in G$ will have closure a maximal torus.

Proposition 3.3 states that there is a proper closed subvariety $X \subset G^{\mathbb{C}}\times G^{\mathbb{C}}$ so that for any proper algebraic subgroup $H$ containing a maximal torus, $H\times H \subset X$.

Now choose a random pair of elements $(\gamma_1,\gamma_2)\in G\times G$ with respect to a measure absolutely continuous with respect to Haar measure. Then with probability $1$, $\overline{\langle\gamma_i\rangle}$ is a maximal torus, since $\cup_i{X_i(\mathbb{R})}$ has measure $0$. Then if $\langle \gamma_1,\gamma_2\rangle$ is not dense in $G$, then $\overline{\langle \gamma_1,\gamma_2\rangle}=H < G$, where $H$ is closed and contains a maximal rank torus. So $(\gamma_1,\gamma_2)\in X$, again occurring with probability $0$.

I think this gives an outline of a proof under these assumptions.

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