The following is for $\epsilon\geq c\vee 1$ for some random constant. For less than $c$, it is unclear how to modify the argument below. Perhaps some scaling argument can do it. 

Also, the following is proving a much stronger result, than what you ask because I tried to do with local time since fBM doesn't have any Markov property. So a weaker argument should work to give it for all $\epsilon$; any ideas are welcome.

The local time $\ell_{x,t}$ of fractional Brownian motio is continuous eg.["A uniform result for the dimension of fractional Brownian motion level sets"][1] or "On the local time of multifractional Brownian motion"

and from "Occupation time problems for fractional Brownian motion and some other self-similar processes"

[![enter image description here][3]][3]

Suppose by symmetry that fBM starts at $x<a$. So consider $A_{r}=(a,a+r)$ and $A_{-r}=(a-r,a)$ and their occupation times during $I=:[\sigma(a),\sigma(a)+\epsilon]$

$$\mu_{I}(A_{r})=\int_{A_{r}}\ell_{I,x}dx, \mu_{I}(A_{-r})=\int_{A_{r}}\ell_{I,x}dx.$$

Now by contradiction suppose that only one of those occupation time is zero, say $ \mu_{I}(A_{-r})=0$. Both occupation measures cannot be both zero because that would imply that fBM either jumps (but it is continuous) or it is constant equal to $a$ but in fact it is nowhere differentiable; this can be proved with local time continuity as in [here][4]. So we get $ \mu_{I}(A_{r})>0$.

However, we use bound $E[\ell_{x,I}]\geq E[\ell_{x+r,I}]-c|I|^{1-H(1+\delta)}r^{\delta}$ to get 

$$0= E[\mu_{I}(A_{-r})]> E[\mu_{I}(A_{r}) -c|I|^{1-H(1+\delta)}r^{\delta+1}].$$

By taking large enough $r$ (eg. $r>c_{Holder}\epsilon^{H-s}$ for $s>0$), we get $ \mu_{I}(A_{r})=|I|=\epsilon>0$ and so

$$0= E[\mu_{I}(A_{-r})]> E[\epsilon-c_{1}\epsilon^{1-H(1+\delta)}r^{\delta+1}].$$

Since $\epsilon>c\vee 1$ for some large random $c$ depending on $c_{Holder},c_{1}$, we get $\epsilon^{H}>r>c_{Holder}\epsilon^{H-s}$ and so 

$$1-c_{1}(\frac{r}{\epsilon^{H}})^{1+\delta}=1-c_{1}(c_{Holder}\epsilon^{-s_{1}})^{1+\delta}>0$$

for $s_{1}<s$.

  [1]: https://arxiv.org/abs/2003.01423
  [2]: https://i.sstatic.net/rG4gr.png
  [3]: https://i.sstatic.net/b609q.png
  [4]: https://math.stackexchange.com/questions/2748504/proof-of-non-differentiability-of-brownian-motion-using-local-times