Homotopically equivalent compact Lie groups are diffeomorphic I have the following conjecture:
Two homotopically equivalent compact Lie groups will be diffeomorphic. It may be necessary to restrict ourselves to only semisimple Lie groups. For simply connected compact Lie groups, such an assertion was known to Toda (1976), he even proved that they will be isomorphic.
Any commentary (for or against)?
 A: The OP seems to be interested in evidence "for or against", so I will give a partial result.
Definition: Let $G$ be a compact connected semisimple Lie group, $\varphi_G:\tilde{G}\to G$ its universal cover, and $\Delta\cong \pi_1(G)$ the Deck group of $\varphi_G$.  There exists simple simply-connected compact groups $G_1,...,G_n$ so $\tilde{G}\cong \prod_{i=1}^nG_i$.  If $\Delta=\prod_{i=1}^n\Delta_i\leq \prod_{i=1}^nZ(G_i)\cong Z(\tilde{G}),$ we will say that $G$ is adjoint-like.
Remark: If $G$ is adjoint-like then it is necessarily a cartesian product of simple groups.  At one extreme, when $\Delta$ is trivial, it includes the simply-connected case, and at the other extreme when $\Delta$ is the full center, then $G$ is of adjoint-type (centerless); which motivates the name.
Notation: For a Lie group $G$ we will use the notation $G_0$ for its identity component, $DG:=[G,G]$ for its derived subgroup, and $Z(G)$ for its center.
Theorem: Let $G$ and $H$ be homotopic compact Lie groups. If both $DG_0$ and $DH_0$ are adjoint-like, then $G$ and $H$ are diffeomorphic.
Proof: First note that both $G$ and $H$ have a finite number of connected components.  Since $G/G_0\cong \pi_0(G)\cong\pi_0(H)\cong H/H_0$ they have the same number of components.  And since each component of $G$ (resp. $H$) is diffeomorphic to $G_0$ (resp. $H_0$), it follows that $G$ and $H$ are diffeomorphic iff $G_0$ and $H_0$ are diffeomorphic.
Second, by a result of Borel (Proposition 3.1 in Sous-Groupes Commutatifs et Torsion des Groupes de Lie Compacts Connexes, 1960), $G_0$ is diffeomorphic to $DG_0\times Z(G_0)_0$ and likewise $H_0$ is diffeomorphic to $DH_0\times Z(H_0)_0$.  In each case the identity component of the center is a torus (a product of circles).  Since $G$ and $H$ are homotopic their identity components are as well, and in particular, the fundamental groups are isomorphic.  Consequently, since the fundamental group of a semisimple Lie group is finite, the ranks of the central tori $Z(G_0)_0$ and $Z(H_0)_0$ can be determined from the corresponding fundamental groups.  Thus, $G_0$ and $H_0$ are diffeomorphic iff $DG_0$ is diffeomorphic to $DH_0$.
Lastly,since $G_0$ and $H_0$ are homotopic, and the central tori $Z(G_0)_0$ and $Z(H_0)_0$ are diffeomorphic (from the previous step), we conclude that $DH_0$ and $DG_0$ are homotopic as well.  From Theorem 2 in Compact Lie Groups with isomorphic Homotopy Groups (1998) by Boekholt, we have that $DH_0$ and $DG_0$ are locally isomorphic and hence their universal covering spaces $\tilde{DH_0}$ and $\tilde{DG_0}$ are diffeomorphic and their deck groups are abstractly isomorphic.  However, since $DH_0$ and $DG_0$ are both adjoint-like, they are cartesian products of simple groups, arising from the quotient of corresponding simple factors in $\tilde{DH_0}$ and $\tilde{DG_0}$ by central subgroups of those simple factors.  We conclude that each pair of corresponding simple factors are homotopic and hence by Theorem 9.3 in The Cohomology of Quotients of Classical Groups by Baum & Browder (1963) each pair of corresponding simple factors are diffeomorphic.
The result follows. $\Box$
Corollary:  If $G$ and $H$ are homotopic semisimple compact Lie groups that are simply-connected or adjoint-type, then they are diffeomorphic.
Remark: There is some tautological reasoning with this corollary since the simply-connected case is Toda's 1976 theorem in A note on compact semi-simple Lie groups, and I think it was used in one of the references I quote above (but there is nothing inconsistent).  But as far as I know the adjoint-type case seems new (although not by much).  Also, I am being slightly sloppy with the difference between "homeomorphism" and "diffeomorphism" since they are equivalent here, and similarly I am being sloppy with the difference between "weak homotopy equivalence" and "homotopic" since compact Lie groups are homotopic to CW complexes and Whitehead's theorm.
Remark: The analysis shows exactly how a counter-example to the general question could arise, if it exists.  One needs to look at simply-connected semisimple groups with at least two factors and quotient by isomorphic central subgroups that do not (both) arise as a product of the centers of the simple factors.  Any two groups constructed this way will be homotopic, and conversely the above proof shows that any two such groups that are homotopic must arise this way if they have any chance of not being diffeomorphic.
