Positive eigenvalues and two-letter generalized words

A generalized word in two letters $A$ and $B$ is an expression of the form $W=A^{{\alpha }_1}{B}^{{\beta }_1}{A}^{{\alpha }_2}{B}^{{\beta }_2}\mathrm{.}\mathrm{.}\mathrm{.}{A}^{{\alpha }_N}{B}^{{\beta }_N}$ in which the exponents ${\alpha }_i$, ${\beta }_i$ are nonzero real numbers. When independent positive definite matrices are substituted for $A$ and $B$, we are interested in whether $W$ necessarily has positive eigenvalues. This is known to be the case when N=1 and has been studied in case all exponents are positive by two of the authors. When the exponent signs are mixed, however, the situation is quite different (even for 2-by-2 matrices), and this is the focus of the present work.