A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s
$2$
-variable
$p$
-adic
$L$
-functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a
$2$
-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field
$K$
(where an odd prime
$p$
splits) of an elliptic curve
$E$
, defined over
$\mathbb{Q}$
, with good supersingular reduction at
$p$
. On the analytic side, we consider eight pairs of
$2$
-variable
$p$
-adic
$L$
-functions in this setup (four of the
$2$
-variable
$p$
-adic
$L$
-functions have been constructed by Loeffler and a fifth
$2$
-variable
$p$
-adic
$L$
-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the
$\mathbb{Z}_{p}^{2}$
-extension of
$K$
. We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.