We consider the simple random walk on the Euclidean lattice in transient dimensions. It is known that the number of distinct visited sites is asymptotically linear in time. The probability of visiting a fewer number of sites (with a difference of the order of the mean) was evaluated asymptotically by Phetpradap in 2010, taking up the seminal work of van den Berg, Bolthausen and den Hollander in 2001 (concerning the volume of a Wiener sausage). In a work in progress with Dirk Erhard (Salvador de Bahia, Brazil) we consider the random walk conditioned to such a rare event and prove that the occupation measure of a certain skeleton of the random walk converges (for large enough deviations) to a unique optimal profile. Our proof of this so-called tube property makes use of the recent compactification of the space of measures introduced by Mukherjee and Varadhan, and is a first step in the rigourous proof of the Swiss cheese picture proposed by van den Berg, Bolthausen and den Hollander.