How far actually is the Galactic Center IRS 13E3 from Sagittarius A^(ast)?

The Galactic Center IRS 13E cluster is a very intriguing IR object located at $\sim0.13$ pc from Sagittarius A$^\ast$ (Sgr A$^\ast$) in projection distance. There are both arguments for and against the hypothesis that a dark mass like an intermediate mass black hole (IMBH) exists in the cluster. Recently we have detected the rotating ionized gas ring around IRS 13E3, which belongs to the cluster, in the H30$\alpha$ recombination line using ALMA. The enclosed mass is derived to be $M_{\mathrm{encl.}}\simeq2\times10^4$ $M_\odot$, which agrees with an IMBH and is barely less than the astrometric upper limit mass of the IMBH around Sgr A$^\ast$. Because the limit mass depends on the true three-deminsional (3D) distance from Sgr A$^\ast$, it is very important to determine it observationally. However, the 3D distance is indefinite because it is hard to determine the line-of-sight (LOS) distance by usual methods. We would attempt to estimate the LOS distance by spectroscopic informations. The CH$_3$OH molecule is easily destroyed by cosmic ray around Sgr A$^{\ast}$. However, we detected a highly excited CH$_3$OH emission line in the ionized gas stream associated with IRS 13E3. This indicates that IRS 13E3 is located at $r\gtrsim 0.4$ pc from Sgr A$^{\ast}$.