  The first half of Onsager's conjecture states that the Euler equations of an
ideal incompressible fluid conserve energy if $u (\cdot ,t) \in C^{0, \theta}
(\mathbb{T}^3)$ with $\theta > \frac{1}{3}$. In this paper, we prove an
analogue of Onsager's conjecture for several subgrid scale $\alpha$-models of
turbulence. In particular we find the required H\"older regularity of the
solutions that ensures the conservation of energy-like quantities (either the
$H^1 (\mathbb{T}^3)$ or $L^2 (\mathbb{T}^3)$ norms) for these models. We
establish such results for the Leray-$\alpha$ model, the Euler-$\alpha$
equations (also known as the inviscid Camassa-Holm equations or Lagrangian
averaged Euler equations), the modified Leray-$\alpha$ model, the
Clark-$\alpha$ model and finally the magnetohydrodynamic Leray-$\alpha$ model.
In a sense, all these models are inviscid regularisations of the Euler
equations; and formally converge to the Euler equations as the regularisation
length scale $\alpha \rightarrow 0^+$. Different H\"older exponents, smaller
than $1/3$, are found for the regularity of solutions of these models (they are
also formulated in terms of Besov and Sobolev spaces) that guarantee the
conservation of the corresponding energy-like quantity. This is expected due to
the smoother nonlinearity compared to the Euler equations. These results form a
contrast to the universality of the $1/3$ Onsager exponent found for general
systems of conservation laws by (Gwiazda et al., 2018; Bardos et al., 2019).
