The key theorem here is the following:

Theorem (Kato, 2004): For any $E$ and any $p$, the "fine Selmer group" $Sel_p^0(E / \mathbb{Q}_\infty) = \operatorname{ker}\Big(Sel_p(E / \mathbb{Q}_\infty) \to H^1(\mathbb{Q}_{p, \infty}, E[p^\infty])\Big)$ is cotorsion.

One of the great things about this theorem is that both its statement and its proof are completely independent of the local behaviour of $E$. The dependence on local behaviour comes when you try to use this to deduce things about the classical Selmer group $Sel_p(E / \mathbb{Q}_\infty)$ from Kato's theorem.

Combining Kato's theorem, Poitou--Tate duality, and a theorem in local Iwasawa theory due to Berger, one gets the following consequence:

Corollary: The corank of $Sel_p(E / \mathbb{Q}_\infty)$ is 1 if $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible, and 0 otherwise.

So one needs to check that $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible if and only if $E$ has potentially supersingular reduction, which is a fun exercise.

Yes, the corank conjecture is a theorem for elliptic curves over $\mathbb{Q}$. The key to the proof is the following:

Theorem (Kato, 2004): For any $E$ and any $p$, the "fine Selmer group" $Sel_p^0(E / \mathbb{Q}_\infty) = \operatorname{ker}\Big(Sel_p(E / \mathbb{Q}_\infty) \to H^1(\mathbb{Q}_{p, \infty}, E[p^\infty])\Big)$ is cotorsion.

One of the great things about this theorem is that both its statement and its proof are completely independent of the local behaviour of $E$. The dependence on local behaviour comes when you try to use this to deduce things about the classical Selmer group $Sel_p(E / \mathbb{Q}_\infty)$ from Kato's theorem.

Combining Kato's theorem, Poitou--Tate duality, and a theorem in local Iwasawa theory due to Berger, one gets the following consequence:

Corollary: The corank of $Sel_p(E / \mathbb{Q}_\infty)$ is 1 if $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible, and 0 otherwise.

So one needs to check that $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible if and only if $E$ has potentially supersingular reduction, which is a fun exercise.

The key theorem here is the following:

Theorem (Kato, 2004): For any $E$ and any $p$, the "fine Selmer group" $Sel_p^0(E / \mathbb{Q}_\infty) = \operatorname{ker}\Big(Sel_p(E / \mathbb{Q}_\infty) \to H^1(\mathbb{Q}_{p, \infty}, E[p^\infty])\Big)$ is cotorsion.

One of the great things about this theorem is that both its statement and its proof are completely independent of the local behaviour of $E$. The dependence on local behaviour comes when you try to use this to deduce things about the classical Selmer group $Sel_p(E / \mathbb{Q}_\infty)$ from Kato's theorem.

If I'm not mistaken, combining Kato's theorem, Poitou--Tate duality, and a theorem in local Iwasawa theory due to Berger, one gets the following consequence:

Corollary: The corank of $Sel_p(E / \mathbb{Q}_\infty)$ is 1 if $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible, and 0 otherwise.

So one needs to check that $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible if and only if $E$ has potentially supersingular reduction. If $E$ has good ordinary, or bad multiplicative, reduction, then $T_p(E) |_{G_{\mathbb{Q}_p}}$ is reducible, and if $E$ has good supersingular reduction it is irreducible; so those cases are OK. I expect the additive-reduction cases should also work -- this should be a fun exercise to check.

The key theorem here is the following:

Theorem (Kato, 2004): For any $E$ and any $p$, the "fine Selmer group" $Sel_p^0(E / \mathbb{Q}_\infty) = \operatorname{ker}\Big(Sel_p(E / \mathbb{Q}_\infty) \to H^1(\mathbb{Q}_{p, \infty}, E[p^\infty])\Big)$ is cotorsion.

One of the great things about this theorem is that both its statement and its proof are completely independent of the local behaviour of $E$. The dependence on local behaviour comes when you try to use this to deduce things about the classical Selmer group $Sel_p(E / \mathbb{Q}_\infty)$ from Kato's theorem.

Combining Kato's theorem, Poitou--Tate duality, and a theorem in local Iwasawa theory due to Berger, one gets the following consequence:

Corollary: The corank of $Sel_p(E / \mathbb{Q}_\infty)$ is 1 if $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible, and 0 otherwise.

So one needs to check that $T_p(E) |_{G_{\mathbb{Q}_p}}$ is irreducible if and only if $E$ has potentially supersingular reduction, which is a fun exercise.