You can make the cut, as long as you prove the worm has only been in one hemisphere.
First of, the worm cannot possibly have gone through the center (that path would take 70mm, so the center is 'safe'.
The worm's path is most inconvenient if it remains in a plane, as that maximizes it's useful length available to create an 'inconvenient' path for the apple slicer.
Now, assume a plane through the apple and the 'point of entry', and assume the origin in the center of the apple, and the X-axis going through the point of entry. We need to prove that from this situation, there is no way for the worm to create a path that doesn't allow for a suitable cut. The only way the worm can block a such path, is by having an angle of 180 degrees or higher created, compared to the origin, at the side of the worm's path.
(worm's path in red, 'angle' as described shown in green)
Such a part, at it's minimal length, would be $2x$ the radius, or 70mm. We know however that the path is only 69mm. Therefore the worm cannot ensure there is no untouched hemisphere; so the cut is always possible: cut in a plane perpendicular to the displayed plane, which goes through the center of the apple and any line in the 180°+ range that is untouched by the worm.
Given wording is perhaps a bit complex, and as some have asked more detail on why I focussed on paths in a plane, I'll hereby use the wording of Ivan Barreto from the comments:
Another way of stating it is that the projection of the tunnel in the
xy plane should obscure an angle greater than or equal to 180 degrees.
However, any such tunnel is 70mm or longer. Hence, since a curve is no
shorter than its projection, it should also be 70mm or longer