Let see the details of the curve; Let $K = \operatorname{GF}(3^m)$ and the curve be defined by the equation $$E(K):y^2 = x^3 + 2x + 1$$
Yes, it is supersingular
The group of rational points has order $$n = 19088056323407827075424725586944833310200239047$$ The order has two factors; $7 \cdot 2726865189058261010774960798134976187171462721$.
The second factor ( large one) is $\approx$ 150-bit number.
The generic DLog attack requires $\sqrt{n}$-time, so the security of the curve cannot be larger than $2^75$. Therefore cannot be used securely for ECDH.
In today's standards, we at least require 128-bit security. That is why the Curve25519 is preferable, with some other properties like twist security
It has no twist security at all. The twist has an order $19088056323407827075424246988286372075141058881$ and it has two large factors $(9594160501626613625431,1989549405617260510054951)$, (approx each is a 73-bit number) therefore no twist security.
Curve that uses binary extension field $\operatorname{GF}(2^m)$ are effective in the calculation, however, some binary extension has no longer secure effective sizes. Using 3 as a abase field is harder to use a large field like Curve25519.
According to the current NIST curve, it has lower security, though some of them don't twist security.