Recuitment company fee

Query: Recently, I contacted a recruitment company to hire candidates in my office. They sent a few candidates for interview. After conducting the interview with one candidate, I gave her feedback to consultancy that candidate is good but we want to negotiate in salary. The consultancy later informed that candidate will not negotiate. So, the matter was closed. No further discussion with anyone happened after this. After 20 days, the candidate called me directly saying that she discussed the matter with parents, and has found online that I still have vacancy online, so she wants to work at lower salary and willing to join. I agreed and asked her to join. Neither I nor the candidate contacted the recruitment company for discussion. Now, again after 20 days, the consultancy people called up and asked us to pay their fees claiming that interview was conducted through them. Although they neither helped in negotiation nor did they assisted in joining of candidate in my office. The consultancy's contract, is as below: "Terms and Conditions of our service: To serve your organizations by understanding that candidates are not mere resumes and clients are not steeping stones, we wish that you will certainly understand our terms and conditions. Charges For middle level positions: 8.33% (one salary) of annual package up to 6 lakhs CTC. For Higher designation: >6 lakhs CTC we charge 10% of annual package AVP & VP: 12.5 % of annual CTC Warranty In case candidate leaves the job within three (3) Months we will give Replacement for the candidate provided. Key Benefit We levied our charges within 10 Days from the selection date of the candidate. Recruit at your own leisure 24 hrs a day, 7 days a week. Target profiles/Headhunt quality managers. Maintain confidentiality of your organization. Free consultancy/opinion in HR strengthening and development. We also provide motivational training to employees of your organization. We respect and believe that transparency in this regard would work as a key for achieving success. We recruit candidates on the basis of the Job Description and company’s requirement. Further, interview and selection process would be completed by company. We do not take any liability for damage/theft/inappropriate behavior/mistake or any such unfortunate events. After successful recruitment from Venus Consultancy, Surat; company/organization is solely responsible for the risk/benefits from the employee. Waiting for Healthy Business Relations ahead!! " Are we liable to pay the recruitment company ??