We delineate general conditions under which the value of a frequently applied class of singular stochastic control problems of linear diffusions can be represented in a linearized form as an expected supremum of a representing function of the uncontrolled diffusion at an independent exponential random date. We identify the representing function explicitly in terms of known factors from a Volterra integral equation of the first kind. This is done by setting the value accrued from following a standard local time-type reflection policy equal to the expected value of the representing function at the running supremum of the underlying. We also illustrate our findings numerically in two explicitly solvable parameterized models subject to different boundary behaviors.