It has been conjectured by Liero, Mielke, and Savaré that the Hellinger-Kantorovich distance can be expressed as the metric infimal convolution of the Hellinger and the Wasserstein distances. This statement is quite clear, at least intuitively, if one compares the dynamical representations of the three distances. However, no rigorous proof had yet been provided. We first discuss the infimal convolution between two generic distances, highlighting the difficulties that may arise: in particular, finiteness and triangle inequality may fail. We then sketch the proof of the main result. To prove it, we study with the tools of Unbalanced Optimal Transport the so-called Marginal Entropy-Transport problem that arises as a single minimization step in the definition of infimal convolution. This is a joint work with Nicolò De Ponti and Giacomo Enrico Sodini.