  Hanbury-Brown-Twiss (HBT) correlations for charged pions in central Au+Au
collisions at $\sqrt{s_\mathrm{NN}}=2.4 - 7.7~\text{GeV}$ (corresponding to
beam kinetic energies in the fixed target frame from
$E_{\rm{lab}}=1.23~\text{to}~30~\text{GeV/nucleon}$) are calculated using the
UrQMD model with different equations of state. The effects of a phase
transition at high baryon densities are clearly observed in the HBT parameters
that are explored. It is found that the available data on the HBT radii,
$R_{O}/R_{S}$ and $R^{2}_{O}-R^{2}_{S}$, in the investigated energy region
favors a relatively stiff equation of state at low beam energies which then
turns into a soft equation of state at high collision energies consistent with
astrophysical constraints on the high density equation of state of QCD. The
specific effects of two different phase transition scenarios on the
$R_{O}/R_{S}$ and $R^{2}_{O}-R^{2}_{S}$ are investigated. It is found that a
phase transition with a significant softening of the equation of state below 4
times nuclear saturation density can be excluded using HBT data. Our results
highlight that the pion's $R_{O}/R_{S}$ and $R^{2}_{O}-R^{2}_{S}$ are sensitive
to the stiffness of the equation of state, and can be used to constrain and
understand the QCD equation of state in the high baryon density region.
