Usually it is the village office of your area that is assigned the job of verifying and reporting on the matter. The assigned official will make enquiries in your neighbourhood and also with their own dependable sources and if things are found as informed by you, they will report that it was verified and found true and factual and Legal Heirship Certificate may be issued.
Then the Taluk Office will publish a public notice regarding the same in the Gazette ( in the relevant chapter and section) and seek objections if any from public within a specified deadline. After the deadline last date, in next few working days, Taluk office will issue the Legal Heirship Certificate.
Sometimes the publication may be in the next Gazette depending on when the application and report are made.
It is generally acceptable if the time taken is about three months.
In the present case , you may visit the Taluk Office with your reference and get the status. You can see the State Gazette current issue and if notice is not there verify the next issue. Then wait till the last date and approach the Taluk office after a few days. It is better to buy and keep a copy of the issue of the Gazette where the notice is published.