  We study the $L^2$ norm of the Eisenstein series $E(z,1/2+iT)$ restricted to
a segment of a geodesic connecting infinity and an arbitrary real. We
conjecture that on slightly thickened geodesics of this form, the Eisenstein
series satifies restricted QUE. We prove a lower bound that matches this
predicted asymptotic. We also prove an upper bound that nearly matches the
lower bound assuming the Riemann Hypothesis (unconditionally, the sharp upper
bound holds for almost all $T$). Finally, we show the restricted QUE conjecture
for geodesics with rational endpoints.
