This is a sequel to my earlier question, where I asked for an example of a shift space that is mixing but not intrinsically ergodic -- that is, it has multiple measures of maximal entropy (MMEs). Steve Huntsman's answer referred me to a paper by Haydn that gives such an example: however, in that example the two MMEs are supported on disjoint compact subsets of the shift space, and so in some sense the shift can be viewed as two intrinsically ergodic shifts that have been "glued together".

Is there an example of a transitive shift with multiple MMEs that are all fully supported? More precisely, does anybody know of a transitive shift space $X\subset \{0,1,\dots,p-1\}^\mathbb{Z}$ for which there are two distinct ergodic measures $\mu_1, \mu_2$ such that

1. $h_{\mu_1}(\sigma) = h_{\mu_2}(\sigma) = h_\mathrm{top}(X,\sigma)$;
2. $\mu_i(U)>0$ for every open set $U\subset X$ and $i=1,2$?

This is a sequel to my earlier question, where I asked for an example of a shift space that is mixing but not intrinsically ergodic -- that is, it has multiple measures of maximal entropy (MMEs). Steve Huntsman's answer referred me to a paper by Haydn that gives such an example: however, in that example the two MMEs are supported on disjoint compact subsets of the shift space, and so in some sense the shift can be viewed as two intrinsically ergodic shifts that have been "glued together".

Is there an example of a transitive shift with multiple MMEs that are all fully supported? More precisely, does anybody know of a transitive shift space $X\subset \{0,1,\dots,p-1\}^\mathbb{Z}$ for which there are two distinct ergodic measures $\mu_1, \mu_2$ such that

1. $h_{\mu_1}(\sigma) = h_{\mu_2}(\sigma) = h_\mathrm{top}(X,\sigma)$;
2. $\mu_i(U)>0$ for every open set $U\subset X$ and $i=1,2$?