Problems with the interaction between the relayer and the multisig wallet

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The problem was that Defender provides relayer addresses in non-checksummed form.
Safe API requires exactly this address format.
So, if you replace the code with this, then everything works perfectly:

import Web3 from 'web3';
import { ethers, utils } from 'ethers';
import SafeApiKit from '@safe-global/api-kit';
import EthersAdapter from '@safe-global/safe-ethers-lib';
import Safe from '@safe-global/safe-core-sdk';
import { SafeTransactionDataPartial } from '@safe-global/safe-core-sdk-types';
import { DefenderRelayProvider, DefenderRelaySigner } from 'defender-relay-client/lib/ethers';

const txServiceUrl = 'http://localhost:8000/txs/';
const safeAddress = '0x512560bB71b98a60bC7e87719D2B8573C935Ff58';
const tokenAddress = '0xFAA27935213A149c5f81442C30BaaD36b15863A9';
const abi = 'erc-20 token abi...';
const amount = '100'; // 100 erc-20 tokens to burn

const {
    API_KEY,
    API_SECRET,
} = process.env;

// Create defender signer:
const credentials = { apiKey: API_KEY!, apiSecret: API_SECRET! };
const relayerProvider = new DefenderRelayProvider(credentials);
const signerOrProvider = new DefenderRelaySigner(credentials, relayerProvider, {
    speed: 'fast'
});

// Convert address to checksummed:
let senderAddress = await signerOrProvider.getAddress();
if (!Web3.utils.checkAddressChecksum(senderAddress)) {
    senderAddress = Web3.utils.toChecksumAddress(senderAddress);
}

// Create safeService instance:
const ethAdapter = new EthersAdapter({
    ethers,
    signerOrProvider
});
const safeService = new SafeApiKit({
    txServiceUrl,
    ethAdapter
});

// Create transactoin description:
const contractInterface = new utils.Interface(abi);
const data = contractInterface.encodeFunctionData('burn', [ethers.utils.parseEther(amount)]);

const safeTransactionData: SafeTransactionDataPartial = {
    to: tokenAddress,
    value: '0',
    data,
    operation: 0
};

// Create transaction:
const safeSdk = await Safe.create({
    ethAdapter,
    safeAddress
});
const safeTransaction = await safeSdk.createTransaction({ safeTransactionData });

// Try to propose transaction:
const safeTxHash = await safeSdk.getTransactionHash(safeTransaction);
const senderSignature = await safeSdk.signTransactionHash(safeTxHash);
await safeService.proposeTransaction({
    safeAddress,
    safeTransactionData: safeTransaction.data,
    safeTxHash,
    senderAddress,
    senderSignature: senderSignature.data
});

console.log('DONE!!!!');