  We study the total mass of high points in a random model for the Riemann-Zeta
function. We consider the same model as in [8], [2], and build on the
convergence to 'Gaussian' multiplicative chaos proved in [14]. We show that the
total mass of points which are a linear order below the maximum divided by
their expectation converges almost surely to the Gaussian multiplicative chaos
of the approximating Gaussian process times a random function. We use the
second moment method together with a branching approximation to establish this
convergence.
