The key here is the "ground". According to your picture, you can see immediately that there is an electric field from the two rods acting (in the downward direction) on the wire connecting between the sphere and the ground. This electric field will 'kick' electrons in the wire, and generate current. The sphere will get charged after a while (it loses electrons). After the sphere is charged, it has some electrical potential difference from the ground, and hence generate electric field in the opposite direction (in this case, upward). That way, it can become equilibrium (so electrons in the wire experience no electric field).
So, the sphere will get negatively charged, but uniformly.
Ps. If this sphere is not groudedgrounded, nothing happens.