According to the PMBOK Guide, a project manager and project team should select the appropriate work methods and tools based on the project characteristics, such as the product scope, complexity, uncertainty, and delivery approach1. The work methods and tools should also be aligned with the business need and project constraints, such as the budget, schedule, quality, and resources2. Therefore, the result of the discussion should be an agreed-upon work method and tools that should fit the business need and project constraints. References:
* 1: PMBOK Guide, 6th edition, Chapter 4: Project Integration Management, p. 74-75
* 2: PMBOK Guide, 6th edition, Chapter 5: Project Scope Management, p. 131-132

According to the best practices in project management, when facing significant changes that impact project timelines, the project manager should first assess the situation with the project stakeholders. This involves analyzing the new circumstances, evaluating the potential impact on the project delivery, and exploring alternative solutions or adjustments to the project plan. Once a clear assessment is made, the project manager should then communicate the impact to the customer, providing them with the most accurate and updated information. This approach ensures that the customer is informed of the situation based on a thorough analysis and collaborative effort, rather than a hasty, unilateral decision by the project manager.
References:
PMI's Code of Ethics and Professional Conduct emphasizes the importance of responsibility, including making decisions and taking actions based on the best interests of society, public safety, and the environment1.
The PMBOK Guide highlights the need for project managers to engage with stakeholders to understand their needs and expectations, and to manage their engagement throughout the project2.
The PMI Guide to Business Analysis states that business analysts should work with stakeholders to define requirements and shape project outputs3.
1: PMP Examination Content Outline - June 2019 - Project Management Institute 2: A Guide to the Project Management Body of Knowle

The project manager should first try to understand why the key stakeholder is not attending the meetings and providing false information. The project manager should discuss the communication needs and preferences of the key stakeholder and tailor the communication accordingly. This will help to build trust and rapport with the key stakeholder and ensure that they receive accurate and timely information about the project progress.
Creating a new communication artifact, emailing the latest status report, or sharing the meeting recordings may not address the root cause of the problem and may not be effective in engaging the key stakeholder. References: PMBOK Guide, 6th edition, page 524; PMP Exam Content Outline, 2021, page 10.

According to the Project Management Professional (PMP) principles, when there is a discrepancy between the received sample and the expected specifications, the project manager should first refer to the procurement contract. This contract would contain the agreed-upon specifications and would serve as the reference point for any disputes. By doing this, the project manager can ensure that the supplier's claim about the measurement being accurate is indeed correct, or if the project engineer's complaint about the sample not being the correct length is valid. References: PMBOKGuide - Sixth Edition, Section 12.3.3.1: Control Procurements