Professor Atsushi Murase from Kyoto Sangyo University was kind enough to send me the preprint. I am uncertain whether I can post it publicly, but if you write me an email I will send it to you.    
<sub>[email protected]</sub>

[![enter image description here][1]][1] 


Professor Tomonori Moriyama from Osaka University gave me some background information:

> The paper you mentioned  is the master thesis of S-I. Kato under the
> supervision of Prof. N. Iwahori, which is hand-written and written in
> Japanese. I read it in the library of Graduate School of Mathematical
> Sciences of University of Tokyo more than 20 years ago. 
> 
> The basic strategy of proof is more or less the same as T. Shintani's 1976
> paper <A HREF="https://www.jstage.jst.go.jp/article/pjab1945/52/4/52_4_180/_article/-char/ja/">On an explicit formula for class-1 “Whittaker functions” on
 GL$_n$ over $p$-adic fields</A>, but there are much more complicated
> case-by-case arguments.

  [1]: https://i.sstatic.net/iLO891j8.png