A canonical proboscis domain \Omega corresponding to contact angle as introduced \gamma_0 by Fischer and Finn and later studied by Finn and Leise, has the property that a solution of the capillary problem exists in \Omega for contact angle \gamma if and only if | \gamma – \frac{\pi}{2}| < | \gamma_0 – \frac{\pi}{2}| . We show in this paper that every such domain can be modified so as to yield the existence of a bounded solution also at the angle \gamma_0 . The modification can be effected in such a way that for prescribed \epsilon > 0 the solution height must-physically become infinite when | \gamma – \frac{\pi}{2}| > |\gamma_0 – \epsilon – \frac{\pi}{2}| , over a subdomain that includes as large a portion of \Omega as desired.